Begin by distributing the \(\displaystyle 4\) through the group:

\(\displaystyle 3x + 14y + 4(xy) - (4)(4x)\)

Next, perform the multiplications:

\(\displaystyle 3x + 14y + 4xy - 16x\)

Group the like terms:

\(\displaystyle 3x - 16x+ 14y + 4xy\)

Combine like terms:

\(\displaystyle -13x+ 14y + 4xy\)

Rearrange the terms to get the answer as it appears in the answer choices.

Begin by multiplying through by \(\displaystyle -4\):

\(\displaystyle 43x^{2} + 2xy - 3y + (-4)(7xy) - (-4)(22x^{2})\)

Perform the multiplications:

\(\displaystyle 43x^{2} + 2xy - 3y - 28xy - (-88x^{2})\)

The double negation becomes addition:

\(\displaystyle 43x^{2} + 2xy - 3y - 28xy + 88x^{2}\)

Group like terms:

\(\displaystyle 43x^{2}+ 88x^{2} + 2xy - 28xy - 3y\)

Combine like terms:

\(\displaystyle 131x^{2} + 26xy - 3y\)

\(\displaystyle 8(x+7) - 3 (x + 10)\)

\(\displaystyle = 8 \cdot x+ 8 \cdot 7 - 3 \cdot x - 3 \cdot 10\)

\(\displaystyle = 8 x+ 56 - 3x - 30\)

\(\displaystyle = 5x+ 26\)

\(\displaystyle 5-9 = 5 + (-9)= -(9-5) = -4\) in normal arithmetic.

In modulo 11 arithmetic, a negative number has 11 added to it as many times as necessary until a positive sum is reached:

\(\displaystyle -4 + 11 = 7\)

Therefore, 

\(\displaystyle 5-9 \equiv 7 \mod 11\)

By the order of operations, in the absence of grouping symbols, multiplication takes precedence over addition and subtraction. Addition and subtraction are then carried out with equal priority, but from left to right, so the subtraction is performed second and the addition last.

\(\displaystyle 7 + x^{2}\) is \(\displaystyle x^{2}\) added to seven; \(\displaystyle x^{2}\) is the square of a number.

\(\displaystyle 7 + x^{2}\) is subsequently "the square of a number added to seven".

The first step to solving is to convert all the numbers to a common form. We will convert to decimals, since this form is used in the answer choices. 

\(\displaystyle \frac{5}{10}=5\div10=0.20\)

Plug this value into the original equation and solve.

\(\displaystyle 0.4 +0.2-4.67\)

\(\displaystyle 0.6-4.67\)

\(\displaystyle -4.07\)

Our final answer is \(\displaystyle -4.07\).

The easiest way to solve this problem is to pick an odd number (we can use 3 for example), and plug it into each answer choice. 

The only answer choice that gives us an odd result is:

\(\displaystyle R-4\)

\(\displaystyle 3-4=-1\)

As a rule of thumb, an even number subtracted from an odd number will always result in an odd number.

If there were 16 ounces of water to start, this amount decreased to 14 ounces when Philip drank 2 ounces. 

\(\displaystyle 16-2=14\)

With 14 ounces remaining, his brother drank half. This would be equal to 7 ounces.

\(\displaystyle 14\div2=7\)

This left the other half of the water, which was also 7 ounces; thus, 7 ounces in the amount of water that remained.

The first step is to transfer \(\displaystyle 17\) minus two times another number equals \(\displaystyle 3\) into mathematical terms. Doing so gives us:

\(\displaystyle 17-2x=3\)

Next, 17 is subtracted from each side. 

\(\displaystyle -2x=-14\)

Each side is now divided by \(\displaystyle -2\). 

\(\displaystyle x=7\)

Therefore, \(\displaystyle 7\) is the correct answer.